Modified Numerov's method for inverse Sturm-Liouville problems

In this paper, we propose a new modified Numerov's method for recovering from eigenvalues a symmetric potential of a Sturm-Liouville operator with Dirichlet boundary conditions. We use interpolation to refine the mesh sufficiently for Numerov's method to be effective even without the asymptotic correction technique of Andrew and Paine. Accuracy and stability of the method are investigated. Convergence of the method is established. Our method is extended to deal with natural boundary conditions. Numerical experiments confirm its competitiveness.